DOCSLIB.ORG

Hydrogen Atoms in a High-Frequency Laser Field

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Hydrogen Atoms in a High-Frequency Laser Field atoms Review Mini-Review: Hydrogen Atoms in a High-Frequency Laser Field Eugene Oks Physics Department, 380 Duncan Drive, Auburn University, Auburn, AL 36849, USA; [email protected] Received: 9 July 2019; Accepted: 15 August 2019; Published: 19 August 2019 Abstract: Because of the continuing advances in developing lasers in the far-ultraviolet and x-ray ranges, studies of the behavior of atoms under a high-frequency laser field are of theoretical and practical interest. In the present paper, we review various analytical results obtained by the method of separating rapid and slow subsystems for various polarizations of the laser field. Specifically, we review the corresponding analytical results both in terms of the quantum description of the phenomena involved and in terms of the classical description of the phenomena involved. We point out that, for the classical description of hydrogen atoms in a high-frequency laser field, there are interesting celestial analogies. We discuss hidden symmetries of these physical systems, the advantages of this analytical method, and the connection between these results and the transition to chaos. Keywords: atoms in a high-frequency laser field; separation of rapid and slow subsystems; celestial analogies; hidden symmetries 1. Introduction Because of the continuing advances in developing lasers in the far-ultraviolet and x-ray ranges, studies of the behavior of atoms under a high-frequency laser field are of theoretical and practical interest. The “high-frequency” means that the laser frequency ! is much greater than any of the atomic transition frequencies !n’n: ! >> !n’n (1) In the present paper we review various analytical results obtained by the method of separating rapid and slow subsystems. Specifically, in Section2 we review the corresponding analytical results in frames of the quantum description of the phenomena involved. In Section2 we follow papers [ 1,2]. In Section3 we review the corresponding analytical results in frames of the classical description of the phenomena involved. In Section3 we follow papers [ 3,4]. In Conclusions, we discuss the advantages of applying the method of separating rapid and slow subsystems to the physical systems under consideration. We emphasize the rich physics behind the corresponding phenomena. 2. Quantum Hydrogen Atoms in a High-Frequency Laser Field It is well-known [5,6] that, for quantum systems in a monochromatic field, it is convenient to use the formalism of quasienergy states. The problem of finding such states of the hydrogen-like atom/ion in a linearly polarized high-frequency laser field was first considered by Ritus [5]. He found quasienergies for the states of the principal quantum number n = 1 and n = 2, for which the perturbation operator U is diagonal in the basis of the spherical wave functions (i.e., the wave functions of the unperturbed atom in the spherical quantization). Papers [7,8] stated, without any proof, that for states of n > 2, the perturbation operator U couples the substates of l and l 2, where l is the orbital momentum quantum number. In paper [8], the study ± of the quasienergies for n > 2 was based on the approximate analogy with the problem of finding Atoms 2019, 7, 83; doi:10.3390/atoms7030083 www.mdpi.com/journal/atoms Atoms 2019, 7, 83 2 of 15 + + energies for the hydrogen molecular ion H2 . Based on the energies for H2 from paper [9], for n > 2, the author of paper [8] found “corrections” to the quasienergies from paper [5] due to the coupling of the substates of l and l 2. However, in a later paper [10], where the authors used the dependence of + ± the energies for H2 on the internuclear distance R from paper [11], they obtained the result that in the limit R coincides with the quasienergies from paper [5], which contradicts paper [8]. !1 Thus, it remained unclear whether the perturbation U couples the substates of l and l 2 at fixed ± n. To answer this question, one should have directly calculated matrix elements of the perturbation between the substates of l and l 2 at fixed n. This was accomplished in paper [1], where the authors ± demonstrated that these matrix elements are zeros. This meant that, in reality, the perturbation operator U does not couple the substates of l and l 2 for any n, so that the expression for quasienergies from ± paper [5] is not limited by n = 1 and n = 2 (as asserted in paper [5]), but is in fact valid for any n. The diagonality of the perturbation operator U in the basis of the spherical wave functions allowed easily calculating the splitting of any spectral line of a hydrogenic atom/ion under the high-frequency laser field. This was the main, fundamental result of paper [1]. Under condition (1), the laser field represents the rapid subsystem, while the hydrogenic atom/ion represents a slow subsystem. The powerful method of separating rapid and slow subsystems allows us to obtain accurate analytical results for such systems. For implementing this method for particular physical systems, there can be some interesting nuances or versions. It is instructive to see how this method was implemented in paper [1], whose results we present below. The Schrödinger equation for ahydrogen-like atom/ion in a laser field (of the amplitude E0) described by the vector-potential A(t) = A sin!t, where A = (0, 0, cE /!), has the following form 0 0 − 0 (here and below the atomic unitsh ¯ = me = e = 1 are used): i@Y/@t = [H + V(t)]Y,H = p2/2 Z/r + A 2/(2c), V(t) = (A p/c)sin!t [A 2/(2c)]cos2!t (2) 0 0 − 0 − 0 − 0 Here, Z is the nuclear charge and r is the distance of the electron from the nucleus. The notation A0p stands for the scalar product (also known as the dot-product) of these two vectors. The term 2 A0 /(2c) is the average vibrational energy of the free electron in the laser field. We seek the solution of Equation (2) in the form Y(t) = exp[ iα(t)] F, α(t) = [A p/(!c)]cos!t [A 2/(8!c2)]sin2!t (3) − 0 − 0 Substituting Equation (3) in Equation (2), we obtain the following equation: i@F/@t = H F,H = exp[iα(t)]H exp[ iα(t)] = H + i[α,H ] + (i2/2)[α,[α,H ]] + ::: = H + H (4) 1 1 0 − 0 0 0 1,stat 1,osc In Equation (4), [α,H0] and [α,[α,H0]] are commutators. H1,stat and H1,osc are the stationary (i.e., averaged over the laser field period 2π/!) and oscillatory parts of the Hamiltonian H1. Under the condition (1), i.e., since the laser field is the rapid subsystem, the primary contribution to the solution of Equation (4) originates from the stationary part H1,stat. According to the second line of Equation (4), H1,stat can be represented in the following form: H = H + γV + γ2V + ::: , γ = E 2/(2!2)2,V = Z(1 3cos2θ)/r3,V = 3Z( 3 + 30cos2θ 35cos4θ)/(4r5) (5) 1,stat 0 1 2 0 1 − 2 − − where θ is the polar angle of the atomic electron, the z-axis being parallel to the laser field. Assuming γ << 1, we will use the perturbation theory for finding the eigenvalues and the eigenfunctions of the Hamiltonian H1,stat. It is important to emphasize the following counterintuitive fact: since the unperturbed system is degenerate, then according to paper [12], the linear (with respect to γ) corrections to the eigenfunctions will originate not only from the term γV1 but also from the 2 term γ V2. Atoms 2019, 7, 83 3 of 15 It is easy to see that the radial part of the matrix element <nlm|V |nl’m>, where l’ = l 2, reduces 1 − to the following type of the integral: Z1 r r p J = z exp( z)Q + (z)Q (z)dz, (s > 0), (6) − k s k 0 m where Qn (z) are the Laguerre polynomials. According to the textbook [13], for <nlm|V1|nl’m> with l’ = l 2 one gets J = 0, so that <nlm|V |nl’m> = 0. This means that the spherical eigenfunctions ' − 1 nlm of the unperturbed Hamiltonian H0 turn out to be the correct eigenfunctions of the zeroth order of the truncated perturbed Hamiltonian H0 + γV1. Therefore, according to paper [12], the eigenvalues Fnλ and the eigenfunctions χnλ of the Hamiltonian H1,stat within the accuracy of the terms ~γ are expressed as follows: (0) (0) 2 2 2 2 P Fnλ = En + γ < nλ V1 nλ >,En = Z /(2n ) + E0 /(2!) , χnλ = 'nλ + γ j j − j (7) (0) (0) < j V nλ > ' /(En E )+ j 1j j − j P P (0) (0) +γ < nµ V1 j >< j V1 nλ > 'nµ/[(En Ej )(< nλ V1 nλ > < nµ V1 nµ >)] µ λf j j j j j − j j − j j , o (8) ++ < nµ V nλ > 'nµ/(< nλ V nλ > < nµ V nµ >) j 1j j 1j − j 1j In Equation (8), λ = (l, m), µ = (l”, m”), j = (n’, l’, m’), n’ , n. Substituting V1 from Equation (5) in Equation (8), we obtain the following for l > 0: F = Z2/(2n2) + E 2/(2!)2 + (Z4E 2/!4)[3m2 l(l + 1)]/[n3(2l + 3)(l + 1)(2l + 1)l(2l 1)] (9) nlm − 0 0 − − For finding Fnlm (i.e., for l = 0), instead of the formula for V1 from Equation (5), we use the following expression: 2 2 " 3 V = Z[1 3cos θ + "(4cos θ 1)]r − ,|"| << 1 (10) 1 − − The expression for V1 from Equation (10) corresponds to the quasi-Coulomb nuclear potential Zr" 1.
Load more

Recommended publications
  • Magnetism, Angular Momentum, and Spin
    Chapter 19 Magnetism, Angular Momentum, and Spin P. J. Grandinetti Chem. 4300 P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin In 1820 Hans Christian Ørsted discovered that electric current produces a magnetic field that deflects compass needle from magnetic north, establishing first direct connection between fields of electricity and magnetism. P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Biot-Savart Law Jean-Baptiste Biot and Félix Savart worked out that magnetic field, B⃗, produced at distance r away from section of wire of length dl carrying steady current I is 휇 I d⃗l × ⃗r dB⃗ = 0 Biot-Savart law 4휋 r3 Direction of magnetic field vector is given by “right-hand” rule: if you point thumb of your right hand along direction of current then your fingers will curl in direction of magnetic field. current P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Microscopic Origins of Magnetism Shortly after Biot and Savart, Ampére suggested that magnetism in matter arises from a multitude of ring currents circulating at atomic and molecular scale. André-Marie Ampére 1775 - 1836 P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Magnetic dipole moment from current loop Current flowing in flat loop of wire with area A will generate magnetic field magnetic which, at distance much larger than radius, r, appears identical to field dipole produced by point magnetic dipole with strength of radius 휇 = | ⃗휇| = I ⋅ A current Example What is magnetic dipole moment induced by e* in circular orbit of radius r with linear velocity v? * 휋 Solution: For e with linear velocity of v the time for one orbit is torbit = 2 r_v.
    [Show full text]
  • Importance of Hydrogen Atom
    Hydrogen Atom Dragica Vasileska Arizona State University Importance of Hydrogen Atom • Hydrogen is the simplest atom • The quantum numbers used to characterize the allowed states of hydrogen can also be used to describe (approximately) the allowed states of more complex atoms – This enables us to understand the periodic table • The hydrogen atom is an ideal system for performing precise comparisons of theory and experiment – Also for improving our understanding of atomic structure • Much of what we know about the hydrogen atom can be extended to other single-electron ions – For example, He+ and Li2+ Early Models of the Atom • ’ J.J. Thomson s model of the atom – A volume of positive charge – Electrons embedded throughout the volume • ’ A change from Newton s model of the atom as a tiny, hard, indestructible sphere “ ” watermelon model Experimental tests Expect: 1. Mostly small angle scattering 2. No backward scattering events Results: 1. Mostly small scattering events 2. Several backward scatterings!!! Early Models of the Atom • ’ Rutherford s model – Planetary model – Based on results of thin foil experiments – Positive charge is concentrated in the center of the atom, called the nucleus – Electrons orbit the nucleus like planets orbit the sun ’ Problem: Rutherford s model “ ” ’ × – The size of the atom in Rutherford s model is about 1.0 10 10 m. (a) Determine the attractive electrical force between an electron and a proton separated by this distance. (b) Determine (in eV) the electrical potential energy of the atom. “ ” ’ × – The size of the atom in Rutherford s model is about 1.0 10 10 m. (a) Determine the attractive electrical force between an electron and a proton separated by this distance.
    [Show full text]
  • 33-234 Quantum Physics Spring 2015
    33-234 Quantum Physics Spring 2015 Derivation of Angular Momentum Rules in Quantum Mechanics R.M. Suter March 23, 2015 These notes follow the derivation in the text but provide some additional details and alternate explanations. Comments are welcome. For 33-234, you will not be responsible for the details of the derivations given here. We have covered the material necessary to follow the derivations so these details are presented for the ambitious and curious. All students should, however, be sure to understand the results as displayed on page 6 and following as well as the basic commutation properties given on this and the next page. Classical angular momentum. The classical angular momentum of a point particle is computed as L = R p, (1) × where R is the position, p is the linear momentum. For circular motion, this reduces to 2 2πR L = Rp = Rmv = ωmR using v = T = ωR. In the absence of external torques, L is a conserved quantity; in other words, it is constant in magnitude and direction. In the presence of a torque, τ (a vector quantity), generated by a force, F, τ = R F, the equation dL × of motion is dt = τ. In a composite system, the total angular momentum is computed as a vector sum: LT = Pi Li. We would like to determine the stationary states of angular momentum and determine the “good” quantum numbers that describe these states. For a particle bound by some potential in three dimensions, we want to know the set of physical quantities that can be used to specify the stable states or eigenstates.
    [Show full text]
  • What Are the Quantum Numbers for the Ground State of the Hydrogen Atom?
    Welcome back to PHY 3305 Today’s Lecture: Hydrogen Atom Part I John von Neumann 1903 - 1957 Physics 3305 - Modern Physics Professor Jodi Cooley One-Dimensional Atom To analyze the hydrogen atom, we must solve the Schrodinger equation for the Coulomb potential energy of the proton and electron. e2 U(r)=− 4πϵ0r The Schrödinger equation will look like - 2 2 2 − ¯h d ψ(x) − e 2 ψ(x)=Eψ(x) 2m dx 4πϵ0x The wave function solving this equation must meet two criteria: 1) ψ(x) must fall to zero as x approaches infinity 2) ψ(x) must be zero at x = 0, so the LHS remains finite at zero. Physics 3305 - Modern Physics Professor Jodi Cooley Any ideas on the solution? −bx ψ(x)=Axe where me2 1 b = = 2 a0 = Bohr Radius 4πϵ0¯h a0 h2b2 me4 E −¯ − 1 = = 2 2 2 2m 2(4πϵ0) ¯h n n = 1, 2, 3, .... Physics 3305 - Modern Physics Professor Jodi Cooley How do we find the constant A? A handy integral - Substituting Physics 3305 - Modern Physics Professor Jodi Cooley Angular Momentum of Classical Orbits Classically, the angular momentum of a planet is given by The direction of the angular momentum is perpendicular to the plane of the orbit. The energy and angular momentum remains constant. In this example, the planetary orbits have the same energy but differing angular momentum. To describe the angular momentum vector, we need three numbers (Lx, Ly, Lz). Physics 3305 - Modern Physics Professor Jodi Cooley Angular Momentum in Quantum Mechanics The angular momentum properties of a 3-D wave function are described by two quantum numbers.
    [Show full text]
  • Angular Momentum 1 Angular Momentum in Quantum Mechanics
    J. Broida UCSD Fall 2009 Phys 130B QM II Angular Momentum 1 Angular momentum in Quantum Mechanics As is the case with most operators in quantum mechanics, we start from the clas- sical definition and make the transition to quantum mechanical operators via the standard substitution x x and p i~∇. Be aware that I will not distinguish a classical quantity such→ as x from the→− corresponding quantum mechanical operator x. One frequently sees a new notation such as ˆx used to denote the operator, but for the most part I will take it as clear from the context what is meant. I will also generally use x and r interchangeably; sometimes I feel that one is preferable over the other for clarity purposes. Classically, angular momentum is defined by L = r p . × Since in QM we have [xi,pj ]= i~δij it follows that [Li,Lj] = 0. To find out just what this commutation relation is, first recall that components6 of the vector cross product can be written (see the handout Supplementary Notes on Mathematics) (a b) = ε a b . × i ijk j k Here I am using a sloppy summation convention where repeated indices are summed over even if they are both in the lower position, but this is standard when it comes to angular momentum. The Levi-Civita permutation symbol has the extremely useful property that ε ε = δ δ δ δ . ijk klm il jm − im jl Also recall the elementary commutator identities [ab,c]= a[b,c] + [a,c]b and [a,bc]= b[a,c] + [a,b]c .
    [Show full text]
  • Understanding Angular Momentum in Hadrons
    Understanding Angular Momentum in Hadrons Kunal Kathuria Charlottesville, VA A Thesis Presented to the Graduate Faculty of the University of Virginia in Candidacy for the Degree of Doctor of Philosophy Department of Physics UNIVERSITY OF VIRGINIA August, 2013 Declaration of Authorship I, Kunal Kathuria, declare that this thesis titled, `THESIS TITLE' and the work pre- sented in it are my own. I confirm that: This work was done wholly or mainly while in candidature for a research degree at this University. Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. Where I have consulted the published work of others, this is always clearly at- tributed. Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is my own work. I have acknowledged all main sources of help. Signed: Date: i "Humility means not being anxious to be honored by others." \Eloquence is truth concisely stated." [AC Bhaktivedanta Swami] If you can't explain it to a six year old, you don't understand it yourself. [Albert Einstein] At the heart of all knowledge lies simplicity of expression. UNIVERSITY OF VIRGINIA Abstract Department of Physics Doctor of Philosophy by Kunal Kathuria Charlottesville, VA The spin puzzle has been a relatively long-standing unresolved issue in hadron physics. We review and derive, using a wave-packet formalism, the first step preceding all spin sum rules: the connection of the angular momentum operator to the gravitomagnetic form factors of the energy-momentum tensor (EMT).
    [Show full text]
  • 117265A0.Pdf
    FEBRUARY 20, I926] NATURE from the nucleus, so that the screening of the nuclear corresponding to the revolution of the electron in an charge by the other electrons in the atom will have orbit large compared with its own size. On this different effects. This screening effect will, however, assumption the spin axis of an electron not affected be the same for a pair of levels which have the same by other forces would precess with a frequency K but different ]'s and correspond to the same different from the Larmor rotation. It is easily orbital shape. Such pairs of levels were, on the older shown that the resultant motion of the atom for theory, labelled with values of k differing by one unit, magnetic fields of small intensity will be of just the and it was quite impossible to understand why these type revealed by Lande's analysis. If the field is so so-called " relativity" doublets should appear separ­ strong that its influence on the precession of the spin ately from the screening doublets. On our view, the axis is comparable with that due to the orbital motion doublets in question may more properly be termed in the atom, this motion will be changed in a way " spin " doublets, since the sole reason for their which directly explains the gradual transformation appearance is the difference in orientation of the of the multiplet structure for increasing fields known spin axis relative to the orbital plane. It should be as the Paschen-Back effect. emphasised that our interpretation is in complete It seems possible on these lines to develop a quanti­ accordance with the correspondence principle as tative theory of the Zeeman effect, if it is assumed regards the rules of combination of X-ray levels.
    [Show full text]
  • Spin-Orbit Interaction and LS Coupling - Fine Structure - Hund’S Rules - Magnetic Susceptibilities
    Luigi Paolasini [email protected] LECTURE 2: “LONELY ATOMS” - Systems of electrons - Spin-orbit interaction and LS coupling - Fine structure - Hund’s rules - Magnetic susceptibilities Reference books: - Stephen Blundell: “Magnetism in Condensed Matter”, Oxford Master series in Condensed Matter Physics. L. Paolasini - LECTURES ON MAGNETISM- LECT.2 - Magnetic and orbital moment definitions - Magnetic moment precession in a magnetic field - Quantum mechanics and quantum numbers - Core-electron models, Zeeman splitting and inner quantum numbers - Self rotating electron model: the electron spin - Thomas ½ factor and relativistic spin-orbit coupling - Pauli matrices and Pauli equation Two-component wave function which satisfy the non-relativistic Schrödinger equation Theorem: The magnitude of total spin s=s1+s2 is s, the corresponding wave function ψs(s1z, s2z) is Dirac: ”Nature is not satisfied by a point charge but require a charge with a spin!” L. Paolasini - LECTURES ON MAGNETISM- LECT.2 Quantum Numbers Pauli exclusion principle defines the quantum state of a single electron n = Principal number: Defines the energy difference between shells l = Orbital angular momentum quantum number: range: (0, n-1) magnitude: √l(l+1) ħ ml = component of orbital angular momentum along a fixed axis: range (-l, l) => (2l+1) magnitude: mlħ s = Spin quantum number: defines the spin angular momentum of an electron. magnitude: √s(s+1) ħ = √3 /2 ħ ms= component of the spin angular momentum along a fixed axis: range (-1/2, 1/2) magnitude: msħ=1/2ħ j = l ± s = l ± 1/2 = Total angular momentum mj= Total angular momentum component about a fixed axis: range (-j,j) L.
    [Show full text]
  • 6-4 Spin Orbit Interaction
    Spin-orbit interaction Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Due Date: March 14, 2015) In quantum physics, the spin–orbit interaction is an interaction of a particle's spin with its motion. The first and best known example of this is that spin–orbit interaction causes shifts in an electron's atomic energy levels due to electromagnetic interaction between the electron's spin and the magnetic field generated by the electron's orbit around the nucleus. ((Llewellyn Hilleth Thomas)) http://www.aip.org/history/acap/biographies/bio.jsp?thomasl Llewellyn Hilleth Thomas (21 October 1903 – 20 April 1992) was a British physicist and applied mathematician. He is best known for his contributions to atomic physics, in particular: Thomas precession, a correction to the spin-orbit interaction in quantum mechanics, which takes into account the relativistic time dilation between the electron and the nucleus of an atom. The Thomas–Fermi model, a statistical model of the atom subsequently developed by Dirac and Weizsäcker, which later formed the basis of density functional theory. Thomas collapse - effect in few-body physics, which corresponds to infinite value of the three body binding energy for zero-range potentials. http://en.wikipedia.org/wiki/Llewellyn_Thomas 1. Biot-Savart law The electron has an orbital motion around the nucleus. This also implies that the nucleus has an orbital motion around the electron. The motion of nucleus produces an orbital current. From the Biot-Savart’s law, it generates a magnetic field on the electron. The current I due to the movement of nucleus (charge Ze, e>0) is given by Idl ZevN , dl where v is the velocity of the nucleus and v .
    [Show full text]
  • Chapter 14 Nuclear Magnetic Resonance
    Chapter 14 Nuclear Magnetic Resonance Nuclear magnetic resonance (NMR) is a versatile and highly-sophisticated spectroscopic technique which has been applied to a growing number of diverse applications in science, technology and medicine. This Chapter will consider, for the most part, magnetic resonance involving protons. Magnetic Properties of Nuclei In all our previous work, it has been sucient to treat nuclei as structureless point particles characterized fully by their mass and electric charge. On a more fundamental level, as was discussed in Chap. 1, nuclei are actually composite particles made of nucleons (protons and neutrons) and the nu- cleons themselves are made of quarks. The additional properties of nuclei which will now become relevant are their spin angular momenta and mag- netic moments. Recall that electrons possess an intrinsic or spin angular momentum s which can have just two possible projections along an arbi- 1 trary spacial direction, namely 2 h. Sinceh is the fundamental quantum unit of angular momentum, the electron is classied as a particle of spin one-half. The electron’s spin state is described by the quantum numbers 1 1 s = 2 and ms = 2 . A circulating electric charge produces a magnetic moment proportional to the angular momentum J. Thus = J (1) where the constant of proportionality is known as the magnetogyric ratio. The z-component of has the possible values z = hmJ where mJ = J, J + 1,..., +J (2) determined by space quantization of the angular momentum J. The energy of a magnetic dipole in a magnetic eld B is given by E = B = zB (3) where magnetic eld denes the z-axis.
    [Show full text]
  • Book 1 Introductory Atomic Physics and Quantum Mechanics
    Book 1 Introductory Atomic Physics and Quantum Mechanics Professor C.R. Viswanathan Electrical Engineering Department University of California at Los Angeles Distinguished Professor Emeritus 1 Chapter 1 Introductory Atomic Physics I.1.1 Introduction An understanding of modern solid state theory is possible only after the fundamentals of quantum mechanical principles and the application of these principles to atomic systems are clearly understood. With this in mind, we will proceed to develop the concepts of quantum mechanics and their application to physical systems in Part I of the book. In Chapter I, we will trace the inability of concepts of classical physics to explain some of the experimental observed phenomena, In particular, classical physics proved to be unsuccessful in the following areas: (1) Inability to provide a satisfactory theory of the atom (2) Inability to explain the characteristic spectral lines of the various elements (3) Failure to provide a theory for the observe characteristics of thermal radiation (4) inadequacy to explain the photoelectric phenomenon Planck (thermal radiation), Einstein (Photoelectric effect) and Bohr (atomic model) made some bold assumptions to remove the failures of classical physics. Each made hypothetical assumptions to suit his particular model. There was no justification for these assumptions other than the fact that the model explained the experimental results. In the following paragraphs we will discuss the above failures of classical physics and the models of Planck, Einstein and Bohr. I.1.2 Early Atomic Theories As in other fields, the early Greeks contributed to the modern atomic theory. Democritus (460-370 BC) postulated that “the universe consists of empty space and an almost infinite number of indivisible and invisible particles” which differ in form, position and arrangement.
    [Show full text]
  • The Hydrogen Atom
    Chapter 6 The Hydrogen Atom 6.1 The One-Particle Central-Force Problem Before studying the hydrogen atom, we shall consider the more general problem of a single particle moving under a central force. The results of this section will apply to any central-force problem. Examples are the hydrogen atom (Section 6.5) and the isotropic three-dimensional harmonic oscillator (Prob. 6.3). A central force is one derived from a potential-energy function that is spherically symmetric, which means that it is a function only of the distance of the particle from the origin: V = V r . The relation between force and potential energy is given by (5.31) as (1F2= - ٌV x, y, z = -i 0V 0x - j 0V 0y - k 0V 0z (6.1 The partial derivatives in 1(6.1) can2 be found1 > by2 the chain1 > rule2 [Eqs.1 (5.53)–(5.55)].> 2 Since V in this case is a function of r only, we have 0V 0u r,f = 0 and 0V 0f r,u = 0. Therefore, 1 > 2 1 > 2 0V dV 0r x dV = = (6.2) 0x y,z dr 0x y,z r dr a b a b 0V y dV 0V z dV = , = (6.3) 0y x,z r dr 0z x,y r dr where Eqs. (5.57) and (5.58)a haveb been used. Equationa b (6.1) becomes 1 dV dV r r F = - xi + yj + zk = - (6.4) r dr dr r 1 2 where (5.33) for r was used. The quantity1 r r in (6.4)2 is a unit vector in the radial direc- tion.
    [Show full text]